Anosov flows with smooth foliations and rigidity of geodesic flows on three-dimensional manifolds of negative curvature
- Creators
- Feres, Renato
- Katok, Anatole
Abstract
We consider Anosov flows on a 5-dimensional smooth manifold V that possesses an invariant symplectic form (transverse to the flow) and a smooth invariant probability measure λ. Our main technical result is the following: If the Anosov foliations are C∞, then either (1) the manifold is a transversely locally symmetric space, i.e. there is a flow-invariant C∞ affine connection ∇ on V such that ∇R ≡ 0, where R is the curvature tensor of ∇, and the torsion tensor T only has nonzero component along the flow direction, or (2) its Oseledec decomposition extends to a C∞ splitting of TV (defined everywhere on V) and for any invariant ergodic measure μ, there exists χ_μ > 0 such that the Lyapunov exponents are −2χ_μ, −χ_μ, 0, χ_μ, and 2χ_μ, μ-almost everywhere. As an application, we prove: Given a closed three-dimensional manifold of negative curvature, assume the horospheric foliations of its geodesic flow are C∞. Then, this flow is C∞ conjugate to the geodesic flow on a manifold of constant negative curvature.
Additional Information
© Cambridge University Press 1990. (Received 13 December 1988)Attached Files
Published - AK-Feres1990.pdf
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Additional details
- Eprint ID
- 76219
- Resolver ID
- CaltechAUTHORS:20170408-163655442
- Created
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2018-03-08Created from EPrint's datestamp field
- Updated
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2021-11-15Created from EPrint's last_modified field